Definition 7.13 (Tree)

Let G = (V, E) be a graph with the properties
 1.

One node is distinguished called root.
 2.

Every node c V \ { root } is connected to one
other node p called the parent of c.
 3.

A tree is connected in the sense, that if we start at any node n
other than the root, move to the parent of n,
more to the parent of the parent of n,
and so on, we reach the root.
G is a tree T.
Definition 7.14 (Child)

If p is the parent of node c,
we also say c is a child of p.
A node may have zero or more children,
but every node other the root has exactly one parent.
Definition 7.15 (Ancestor, Descendant)

Let <
>
be a path in a tree T with
= root.
The
nodes
1 i < k, are called an
ancestor of
and the node
, 1 i < k,
a descendant of
node
Definition 7.16 (Siblings)

Nodes that have the same parent are called siblings.
Definition 7.17 (Leaf)

A leaf is a node of a tree T that has no children.
Definition 7.18 (Height and Depth)

In a tree T the height of a node n is the length of a longest path
from n to a leaf.
The height of a tree is the height of the root.
The depth, or level,
of a node n is the length of the path from the root to n.
Definition 7.19 (Labeled Trees)

A labeled tree is a tree in which a label or value is
associated with each node of the tree.
Definition 7.20 (Subtree)

In a tree T, a node n together with all of its
descendants, if any,
is called a subtree of T.
We define binary trees recursivly as follows:
Definition 7.21 (Binary Tree)

The empty tree is a binary tree.
If r is a node and
and
are binary trees,
then is the tree with root r, left subtree
and
right subtree
is a binary tree.
Definition 7.22 (Binary Search Tree)

A binary search tree (BST) is a labeled tree in which
the following property holds at every node x in the tree:
All nodes of the left subtree of x
have labels less than the label of x, and
all nodes of the right subtree of x
have labels greater than the label of x.
The property is called the binary search tree
property.
Examples:


Draw four BST for the nodes: 1, 2, 3, 4, 5, 6
Created by unroff & hptools.
© by HansPeter Bischof. All Rights Reserved (1998).
Last modified: 27/July/98 (12:14)